April 11th, 2017, 10:10 AM  #1  
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208  Probability paradoxes and other difficulties
The following question was quite reasonably asked in another thread but deserves a thread of its own, especially as it is a bit off topic in the original. Quote:
All this question shows is the folly of contemplating the infinite whilst smoking weed. Yes the a priori probability of picking any single number must be zero since there is an infinity of them available. The sample space is an infinite set. So what? So the first mistake is trying to apply the rules from finite sets to infinite ones. Then I must disagree with with the statement that the probability changes to 1. The initial probability following Laplace (determinism) or Venn (frequentism) is the difficulty of the limiting process. But both agree that there is one single probability, which is unaffected ie does not change after the event. Which brings us to Bayes. With the Bayesian approach it is perfectly reasonable to assign an initial zero probability and revise it in the light of subsequent information. However revising it to 1, following a single trial would be excessive.  
April 11th, 2017, 11:53 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,763 Thanks: 905 
another problem when trying to use infinite sets in reality is representation. You say I can choose any of an infinite set. Values must be represented somehow, even if just in the imagination. Suppose I try to choose a number that requires more bits to represent it than there are bits of information in the universe. Yes, this is a pretty big number but not that big... after all there are an infinite number of values larger than it. How might one verify that this was the number you selected? You certainly can't write it down anywhere. If someone asked what number you chose how could you respond? So really we have a distribution on $[0, 2^{\text{# bits in the universe}}]$ and we have no trouble dealing rigorously with this finite distribution Last edited by romsek; April 11th, 2017 at 11:57 AM. 
April 11th, 2017, 04:22 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,151 Thanks: 2390 Math Focus: Mainly analysis and algebra  
April 12th, 2017, 02:47 AM  #4 
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233 
I see this differently. The probability of choosing a number is one since you already decided that you will choose a number. The probability of choosing a specific number like 3 is of course zero. I don't see a paradox , the choice must be made in order for the experiment to go forward. After you complete the experiment and look at what you got , then you can wonder , "what is the probability that I would get this?" In my humble opinion , the OP is mixing up the rules of the experiment with the outcome of the experiment. 
April 13th, 2017, 10:13 PM  #5 
Senior Member Joined: Mar 2017 From: . Posts: 277 Thanks: 5 Math Focus: Number theory 
This is the solution to this problem. The sample space is not infinite. Your mind doesn't consider infinite possibilities. It rather selects from a set of numbers you randomly think of, or just one number you have thought of. So the problem is about the mind. The probability cannot therefore be defined since we don't know you have chosen from how many numbers. i.e., how many numbers you have temporarily thought of and chosen from. Let's say we use a computer instead of the mind. Though impossible, let's say if there was a computer in which you could store all natural numbers. It would be impossible to create an algorithm that would randomly selected just one number. But even if we would create such an algorithm, we would get an error if we execute it. So, it is impossible to store an infinite amount of numbers in a computer in reality, so even computers themselves can't help us. Also it is not possible for our minds to select from an infinite extension of numbers. Last edited by skipjack; April 14th, 2017 at 12:47 AM. 
April 14th, 2017, 02:09 AM  #6  
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208  Quote:
"The probability cannot be defined" When you posed this question, you asserted that the probability was at first zero then 1. But you also applied this to a 'random number' "you are told to randomly choose a number" Can you please tell me how a random number can have a probability of 1? For the number to be random there must be at least one other number available and by your definition of probability 1 excludes all other numbers. Last edited by skipjack; June 8th, 2017 at 08:38 PM.  
April 14th, 2017, 02:43 AM  #7  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,519 Thanks: 506 Math Focus: Yet to find out.  Quote:
It could be modified to be paradoxical though.  
April 14th, 2017, 03:48 AM  #8 
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208 
I named this thread 'and other difficulties' fior a reason and this question is one of the other difficulties that spills over into the Collatz thread. Consider a bag of sweets. I take one at random and eat it. Can I take the last one at random? As the number of sweets diminishes, in a finite bag, the probability of chosing a particular sweet increases from 1/N, where N is the total number of sweets, to 1. But in an infinite bag we are in a limiting situation since we will never run out of sweets. This emphasises a difference between finite and infinite sequences. Laplacian (deterministic) probability is based on a finite sequence. Relativistic probability is based on a limit of an infinite sequence. 
April 14th, 2017, 10:16 AM  #9 
Senior Member Joined: Mar 2017 From: . Posts: 277 Thanks: 5 Math Focus: Number theory 
That is my best shot and I'd like to make it clearer so that it is not confusing or incorrect in any sort of way. When told to randomly choose any number, you are using your mind to select a number. You won't consider an infinite set of numbers. You will only consider the ones you are thinking of or just one that you are thinking of. Hence, the probability of choosing a certain number, is undefined or unknown. We don't know you are selecting from a set of how many numbers. Also it is highly likely you won't select from extremely huge numbers. We limit our choice to numbers that are easy to state or write down. But the probability of choosing a number, is 1 since the event itself must occur. The probability of choosing a specific number is what is unknown. On the part where I have used the example of the computer, the computer wouldn't choose any number, since the probability then would be zero, and hence the event of choosing a number will also not occur and hence will also have a probability of zero. 
April 14th, 2017, 11:28 AM  #10  
Senior Member Joined: Jun 2015 From: England Posts: 740 Thanks: 208  Quote:
I thought I knew what you asked but clearly not, so perhaps the others were right. A conjurer offers a deck of cards and says "Chose a card at random" My friend offers a box of chocolates and says "chose a chocolate, you can have any one in the box." You said "You are told to chose a number at random" You have also told me that the process does not matter to the probability. Yet you now wish to restrict the way I choose a number, without telling me beforehand. Why? Most particularly how can something be random and have a probability of 1? You have completely failed to answer, or even offer any answer to that question I asked. Why not?  

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